Regularity and Sensitivity for McKean-Vlasov Type SPDEs Generated by Stable-like Processes

Abstract

In this paper we study the sensitivity of nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes. By using the method of stochastic characteristics, we transfer these equations to the non-stochastic equations with random coefficients thus making it possible to use the results obtained for nonlinear PDE of McKean-Vlasov type generated by stable-like processes in the previous works. The motivation for studying sensitivity of nonlinear McKean-Vlasov SPDEs arises naturally from the analysis of the mean-field games with common noise.Mathematics Subject Classification (2010): 60H15, 60J60, 91A06, 91A15, 49L20, 82C22

On Mean Field Games with Common Noise based on Stable-Like Processes

“…In this section, we formulate the well-posedness and the sensitivity results of equations (4) or (6), proved in our previous paper [25]. For any measure µ and a vector y the measure µ(.…”

“…As an important ingredient in our proof we use our previous results on the regularity and sensitivity of the nonlinear stochastic differential equations of McKean-Vlasov type generated by stable-like processes [25]. As a by-product of our analysis, we obtain a result of independent interest, not linked with any optimization problem, namely the 1/N -rates of convergence for interacting stable-like processes to the limiting measure-valued stable-like process, Theorem 2 (often interpreted as the 'propagation of chaos' property).…”

“…Namely, according to Lemma 1 of [25], equation (6) rewrites in terms of the measures ζ t = T * (−W t )µ t = µ t (. + σ com W t ) as the following non-stochastic equation with random coefficients…”

“…Then for any given T > 0 the following holds. (i) The Cauchy problem for equation 6is well posed almost surely, that is, for any initial condition Y ∈ M + (R d ) it has the unique bounded nonnegative solution µ t (Y ) such that ζ t = µ t (x + σ com W t ) solves (25). (ii) For all t > 0, ∥ζ t ∥ ≤ ∥Y ∥ and ζ t have densities with respect to Lebesgue measure.…”

“…Let equation (25) be well posed and its solutions ζ t depend smoothly on the initial condition in the sense that the variational derivatives δζ t /δζ 0 (x) and δ 2 ζ/δζ 0 (x)δζ 0 (y) exist as signed measures and are continuous bounded functions of x and y. Then the solutions µ t = ζ t (x − σ com W t ) to equation (6) also depend smoothly on the initial condition µ 0 = ζ 0 and the variational derivatives are given by the formulas…”